KABUL—Suspected Islamic State gunmen wearing police uniforms stormed a private television station in the Afghan capital Tuesday, setting off fighting with government forces that left at least one person dead and forced the Pashto-language station off the air in the latest attack on journalists in the country.

The Taliban, the country’s main insurgency that is dominated by members of the majority ethnic Pashtun community, denied any involvement in the attack on Shamshad TV, even as commandos were combing the halls of the broadcaster...